Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and Invertibility

Abstract

A mixed (Dirichlet-Neumann) boundary value problem (BVP) for the "stationary heat transfer" partial differential equation with variable coefficient is reduced to some systems of nonstandard segregated direct parametrix-based boundary-domain integral equations (BDIEs). The BDIE systems contain integral operators defined on the domain under consideration as well as potential-type and pseudo-differential operators defined on oopen submanifolds of the boundary. It is shown that the BDIS systems are equivalent to the original mixed BVP, and the operators of the BDIE systems are invertible in appropriate Sobolev spaces.